Understanding Mathematics: The Abacus

It has been made out to be a fantastic tool almost like a find of the Millennium. However, that is not
the case. It is mostly a wooden frame with a bunch of beads, and it doesn’t do anything by itself.
Compare that with the modern calculator - you type in 2 multiplicands and then press the x key
and pronto you get the answer.
This is not the case with the Abacus. It doesn’t do anything at all. It’s the human behind it that
is doing all the Arithmetic operations. The Abacus serves somewhat like the equivalent of the
computer rams. It records and hold the answer to intermediate calculations and the final answer
as done by the human.

The beads do not have any intrinsic value. They are assigned values only when in use and according
to the numeric system in play. When used to do calculations in the Decimal System each of the
beads in the lower part of a column has a value of 1 while the beads in the upper part of the column
has a value of 5.

Position of beads when not in-play or at rest.

Position of beads when in-play ( stacked against the beads-stop )

When in play ( as pressed against the beads-stop, or being chain pressed against the beads-stop )
each bead represents value of : (1 or 5) x the column value

Peculiarities of Abacus

Both carry and borrow are done differently on the Abacus. On paper when we perform a carry we do this (see figure1.) Just as shown on the example we would write the carry - a number on the top-left of the next column of numbers. This is not possible with the Abacus and hence we need alternatives.
Likewise borrow is also carried out differently. On paper we do this (see figure 2.) as you can observe on a real Abacus there is no chance for you to write a number on the next column. Another major difference between Abacus and paper addition is that when performing Addition on the Abacus we add 2 numbers at a time and upon completion the beads in-play will be the result of the addition. And, since the numbers on any column can only be maximum 9 any carry or any borrowing will be limited to 10. This process is repeated until all numbers are added up.
This is unlike paper-based Addition where we can be adding several numbers and such additions could result in carry exceeding 10 even in hundreds depending on individual. This is so because we tend to add all numbers in a column and hence the result can be quite large (see figure 3.)

Hereon I will introduce terms in order to shorten sentences and words use and to make communications
easier.

8 beads means 1x 5 and 3 x 1 beads etc - no ambiguities as there is no other way to make this combination and 6,7, 9 beads all have similar meanings
5 beads would imply that as much as possible use 1 x 5 single bead

Performing carry with Abacus

There are 2 possibilities :

1 x 5 + 5 x 1
2 x 5

5 x 1 + 1 x 5 = 1 x 10

Return all the beads on the Unit column and then raise 1 bead on the Tens column to the bead-stop.

Performing Carry method 1

2 x 5 = 1 x 10

This is straightforward. Return the 2 upper beads on Unit column to their resting position then
raise 1 bead on the Tens column.

Performing Carry method 2

Performing borrowing with Abacus
When doing Arithmetic with the Abacus you will frequently encounter the situation where you don’t
have enough beads to allow for the execution of your next step ( can be minus or add ).
eg. 8 + 6 + 4 - 9
8 + 6 + 4 = 18 => you will have 1 x 10 and 3 x 1 & 1 x 5 beads in play on the Abacus which also mean you have used up 3 x 1 + 1 x 5 beads on the Unit Column. Total value of any column is 15 and
since you have used up 8 beads the maximum number, you can deduct further without involving borrowing is 7 and hence you don’t have enough beads to subtract 9 - check it out on your Abacus.
Next step - borrow from the Tens column

Method 1

Imitating the way of paper subtraction, we deploy something - a tack, coin or anything small enough
to be placed on top of the Unit column.

The red bead represents the borrow-10 from the Tens column. Notice that the 1 x 10 bead have been returned to rest. Now the Unit column has a value of 17 and the 10 ( red bead ) can be used to deduct the 9. Result as shown in picture.
Potential problem : once the subtraction is completed the item that we placed must be removed before continuing with next subtraction failing which would result in wrong answer.

Borrow method 1

Method 2

The process for this method is :

use all available beads-in-play to deduct to reduce nine - in the example above I will use
8 beads to reduce 9 to 1
use your fingers to remember the remaining number in this case 1
exchange 1 x 10 bead ( the only one in this example ) for 1 x 5 + 5 x 1 beads
deduct the remaining beads in this case 1 to complete the deduction of 9.

Borrow 2nd method

Method 3

I named it Cross Column Subtraction. This is the preferred method and is supposed to be
the default method. I described the other 2 methods for kids who have problem adjusting
from the paper-based addition carry-and- borrow methods. If that happens either of
method 1 and 2 can be used as an interim method until your child understand how it is
to be done. This method is simple. Using the prevailing example - just cross over from the
Unit column to the Tens column and use the 10-bead to perform the subtraction of the 9.
10 - 9 = 1
Return the 10-bead to rest and push a 1-bead on the Unit column to play. This is all it takes
on Abacus to borrow a 10.

Cross Column Subtraction

The twin to Cross Column Subtraction is of course Cross Column Addition. With Cross Column Addition you stretch across to the adjacent column on the left and perform 10 - n such that 10 - n = the number that you need to add.
For example lets say you are trying to add 8 and 9 ( 8 + 9 ). As soon as you set up 8 on the Abacus you will discover that you do not have enough beads to perform +9. So, you stretch over to the next column ( the Tens column ) and add a 10 ( which is just pushing up 1 1x10 bead ) and then return to the Unit column to deduct a 1. The result is you end up performing +10 - 1 = + 9.

Cross Column Addition

Addition with Abacus

Let’s perform an addition with the following numbers 8 + 5 + 9 + 8
So, we will start the Abacus with the default 8.

Addition with Abacus

A more complicated example

97 + 83 + 56 + 69

See Addition (97+83+56+69) here

Subtraction with Abacus

When performing subtraction with the Abacus we take away active beads and put them to rest. The number of beads taken away will be equal to the value that we want subtracted. The beads are removed from the stacked position at the beads stop and moved back to their resting positions.

Subtraction with Abacus

Multiplication Using the Abacus

There are 3 different ways to do multiplication viz:

the way of my elders - repeated addition
applying the multiplier to each digits of the multiplicand - right to left
(essentially the Long Multiplication but with a variation for carry)
determining the value of the answer digit by digit from right to left
starting from the Unit digit to …..

We will perform each method without the Abacus to show the processes involved.

Method 1

The way of my elders - repeated Addition which what multiplication is about.
Eg my favourite 27 x 17

step 1 : break 27 into 20 + 5 + 2
step 2 : 20 x 17
ignore the 0 and perform 17 + 17= 34
20 x 17 = 340
step 3 : 5 x 17 = 85 ( ½ of 16 & 5 refer to my lessons on Time Table )
340 + 85 = 425
step 4: 2 x 17 = 34
425 + 34 = 459

Method 2

Imitating Long Multiplication but without visible carry ( not possible on an Abacus ) and
summing after each multiplication step.

Method 3

When 2 two digit numbers are multiplied ( the result is usually a 3 digits number but could be a 4 if the last digit causes a carry over to the Thousand ) and written in this form:

2 7

1 7

The value of each of the digits Th H T U can be found from the 2 numbers from the
followings:

Step 1 : determine the Unit digit - in this case 49
Step 2 : as you perform the cross-multiplication the results of the 2 sub-steps
are added to the Unit digit